T E M
T E M
The fame may be alfo found thus; the ratio of the o&avc to the common temperate fifth, deficient by £ of a comma, is 55.79763 to 32.38952. The approximating ratios to which are,
i°. greater than the true 2 : ij 7 : 4* 19 : II, 50 : 29, &c. 2°. lefs than the true 1 : 1, 3 : 2, 5 : 3, 12 : 7, 31 : 18, 205 : 1 19. Where we have the Temperatures of, 1 2, 1 9, 3 1 , and 50 parts, before examined.
And here all ratios greater than the true, ought to be reject- ed; becaufe they give the fifth lefs than true, that is, in this cafe deficient by more than ~ of a comma. If we investigate the approximating ratios to the ratio of the femi-tones major and minor, or 5.19529 to 3,28612, we (hall have the ratios 1 : 1, 2 : 1, 3 : 2, 5 : 3, which rc- fpcdtively give the Temperatures of 12, 19, 31 and 50 parts, before defcribed.
Again, inveftigating the approximating ratios of the fifth to the third major, we fliall find 7:4, 9:5, a : 6, 29 : 16, which will alfo give the Temperatures 12, 19, 31, 50> as before.
Laftly, the approximated ratios of the octave to the true fifth, are 12 : 7 and 53 : 31 greater than the true. The others being of no ufe, fince the fifth mult neceffarily be di- miuifhed. Here we find the Temperature of 53 parts. As to the Temperatures of 43 and 55, being deftituu or' any inulical foundation, it is no wonder tney do not appear by tins method of inveftigation.
Mr. Huygens, in his Cofmutheoros, fays, that the tone or pitch of the voice cannot be preferved, unlefs the confonants be tempered, fo as to deviate a little from the highett perfection. I'or the proof of this ailertion, he brings a melody conlifting of the following founds, C, F, D, G, C; where, if the in- tervals were to be fung periect, by taking the interval from C to F a true fourth afcending, from F to D a thud minor defcending, from D to G a true fourth afcending, and laltly, from G to C a true fifth defcending; we lhould toll a comma below the C from whence we began. Therefore, il; we were to repeat this ferics of notes nine times, we ihould at lalt fall near a tone major below our fiilt found. Mr. Huygens's folution of this difficulty is, that we remember the note from whence we let out, and return to it by a lecret Temperature, thereby finging the intervals a little imperfect; which, he fays, will be found neceflary in almolt all longs or melodies.
A like difficulty is mentioned in the Memoirs of the Royal Academy of Sciences; and is there urged for the neceffity of a Temperature, even for finging in the fame key. And Mr. Huygens's (olution of the difficulty is there approved of. An. 1707. p. 264.
But the folution of thefe learned gentlemen is, as yet, far from being decifivc. No experiment has yet been brought to (hew that the human voice fings tempered notes; not even when accompanied by tempered inftruments. It feems to us, on the contrary, that an exercifed voice guided by a good ear, fings true, even though accompanied by a mif-tuned inft.ru- ment, as harpfichords moll frequently are, efpecially in tranf- pofed keys. And were theie inftruments always as well tuned as art could make them, yet their tones would be equal; and it feems evident to the ear, that the human voice finging na- turally two tones in fucCeffion, as (J, D, E, never makes them equal; and cannot, without great difficulty, and by means of a variation of harmony, be brought to make them equal.
Another folution, therefore, of Mr. Huygens's difficulty, muft be fought for. The truth f»ems to be, that the fecond of the key muft be the true tone-major above the key, and there- fore the third between the fecond and fourth of the key muft be fung deficient by a comma. Thus in the key of C, from C to D will be a tone major = f , and from D so F will be a deficient third = |?. See the article Interval. Mr. Huygens's melody therefore, will ftand thus ■ C, F, D, G, C.
f X if X f X | = i. And the voice would fing the interval F, D, juft as if the note E had been intcrpofed; in which cafe the notes would be C, F, E, D G C.
Thefe notes all come within the diatonic fcale of C; and the voice naturally falls upon the note from whence it fet out. The fame anfwer will hold in the example, mentioned in the Memoirs of the Academy of Sciences; where the intervals b B, G, E, C, occur. And here the interval from b B to G fhould be taken = *£ = £| x to* as in the former example; and for the fame reafon, the key being F. There feems therefore no repugnancy between the practice and theory of mufic, while the melody is confined to one key; but it muft be owned, that in transitions from key to key, efpecially where feveral parts are to make harmony with each other, there ftill remain difficulties, not mentioned by Mr. Huygens, or any other writer we know rjf, which might de- ferve a father examination.
We muft not omit mentioning, that the learned Dr. Smith, in his Harmonics, has not onfy carried the theory of Tempe- raments^ or Temperatures, far beyond all the authors that pre- ceded him; but has (hewn how to tune an inltrument, ac-
cording to any propofed Temperament, "by the car only, which is certainly a molt ingenious difcsvery.
This learned author * prefers what he calls the Temperament of equal harmony, which differs infenlibly from the diWiion of the octave into 50 parts, to ah others; and iniilts, that it labours under the re we It defects, and is of all others the molt agreeable in practice. In the fyftem of equal harmony the Temperaments of the fifth, third major and third minor, are reipedtively, r s „ and t \ and , 3 B of a comma lefs than the truth L '. It would be impoilible here to do juftice to the learned au- thor's rtalbiiiflgs on this luujcct; we (hall only add, that he eltabliifies, contrary to the common opiniun, tlut the lela tem- ple conlonances, generally (peaking, will not bear io great Temperaments as the limpler conlbnances -'. — [■' 6Wr//s ciar- lnonics, p. 172, 188. " (hid. p. 172. c Ibid. p. 146.] Dr. Smith mentions a Temperament communicated to him by the ingenious Mr. Harnion, which contilts in making the proportion between the octave and third major equal to tnat of the circumference of a circle to its diameter. In this lempe- ^rameni the third major is diminifhed by ^ of a comma, but the thud minor is very near the truth, and extremely beauti- ful. Smith, pref. p. xi.
A Jatc autnor- feems to think the divifion of the octave into 31 patts, not to be of modern invention, but neceflanly im- plied in trie doctrine of the antients. At firlt fight it would leem, as if the antients made hut 24 dieiis or divmons m the octave, viz. ten to each fourth, and four to the tone; which (the octave being equal to two fourths and a tonel gives twenty-four diefcb to the octave. But the author jutt quuud contends, that this divifion is to be underltood only in one tendon, that is either afcending or defcending; but that ac- curately (poking, if we coniider all the dieles or divihons of the fourth, both afcending and defcending, we lhall find thir- teen; five to each tone, and three to the ferni-tone major; and confequently thirty-one divihons in the octave. 1 hcie indeed are not all naturally equal; but if we make them fo, we lhall have a Temperature Known by the moderns under the name of Huygens's Temperature ' b . — ■ [ 4 Dr. Pepujeb, in Phil, Tianf. N». 481. p. 273. ° Phil. Tranf. ibid.j oee the article Dii; sis. Temperature, or Climate for plants. The difference of Climate, or Temperature of the air has a very great effect on plants, 'i he different degree of heat is the great caufe of thefe changes, and the different degree of moilture fomewhat aifilts in it. The American and Afian plants, famous in me- dicine when of the growth of their native foils, yet when removed into our Climate, though they grow and even pro- duce their flowers and ripen their feeds, which feems the lait perfection* of a plant, when put to the trial, have been always found to want their proper medicinal virtues. Many of thof'e plants and trees, which though natives of an- other Climate, will endure the open air with us, and grow in our gardens; yet lofe much of their ftrength, and become dwarfs, in proportion to what they were when in their proper Climate. But much lefs violent changes than thefe are able to produce the like effects, at leaft in fume degree. The ie- veral parts of Europe are able to alter the quality of the fame plant, even while it grows naturally in them. Thus the blue aconite or napellus, the root of which is a terrible poifon in the Couth of France; yet in JBritany, a northern province of the fame kingdom, the root of the fame plant, though it feems to grow with equal vigour there, and is equally large and fuc- culent, has no bad effects; but has been eaten by old people and by children, without any injury.
In general, the farther north we go the lefs and lefs hurtful this plant becomes. It is common to almolt all Europe, and we find the inhabitants of fome places dreading it, and that with great juftice, as a fatal poifon; while thofe of others eat the leaves in their fallads, and even efteem them good to re- ftore the appetite.
The common woad which fucceeds well in many parts of England, is not fo certain in France; but the different Cli- mates in different parts of that great kingdom, make ftrange alterations in its juices. In Upper Languedoc they raife great quantities of it, and it makes an extremely fine blue dye for (tuffs of all kinds; but in Britany, though the plant grows as high, and feems to flourifh as well, yet the leaves never are fo fucculent, and the colour obtained from them is not of fo fine a blue, but is dufky and brownilh. This effect of the different Climates, in changing the nature of things produced in them, is not confined to plants; but the animal kingdom fhares in it. The whole ferpent kind are in general larger and more venomous, as we approach the hotter Climates. The tarantula, fo poifonous in the hot coun- tries, is found greatly lefs fo as it is found in more cold re- gions; and the fcorpion, whofe fting is fatal in fome parts of Africa, is little more mifchievous than the wafp or hornet in fome of the coldeft places where it lives. Nay, the Philo- Cophical Tranfafiions inform us, that the bite of the tarantula, even in thofe very places where molt mifchievous, does not exert its power in cold weather; but that a perfon bitten at fuch a time feels not the effe& of the bite till the next fultry hot day, though that may not happen till after two or three weeks,
t The
